Lemma 1 (See Proposition B.8 of [1]}) 

Assume that \(\mathcal {H}\)  does not contain an infinite multiclass Littlestone tree. Then, for any choices of the adversary \(\kappa _1,\ldots ,\kappa _{t-1}\)  up to round \(t\)  and for any choice \(\kappa _t = (\xi _t, y^{(0)}_t, y^{(1)}_t)\)  in round \(t\)  there is a choice \(\eta _t\)  of the learner such that
\(\overline{\mathrm {Ldim}}_k\left(\mathcal {H}_{\xi _1, y_1^{(\eta _1)},\ldots ,\xi _{t-1}, y_{t-1}^{(\eta _{t-1})},\xi _t,y_{t}^{(\eta _{t})}}\right)< \overline{\mathrm {Ldim}}_k\left(\mathcal {H}_{\xi _1, y_1^{(\eta _1)},\ldots ,\xi _{t-1}, y_{t-1}^{(\eta _{t-1})}}\right)\,.\) 

